Intuition The One Core IdeaA circle equation is nothing more than a disguised sentence: "every point that sits exactly r read that sentence whether it's written neatly (standard form) or scrambled up (general form), and to translate between the two without losing the treasure — the centre — hidden inside.
This page assumes you have seen nothing . We are going to build a circle equation one symbol at a time . By the last section every mark you meet in the parent topic will already have a plain meaning and a picture — and each new symbol will lean only on the ones introduced before it.
Even the tiniest marks need a meaning before we lean on them.
Definition Equals, plus, minus, zero
= the same number as the thing on the right". It is a balance , not an instruction to compute.+ add (put together); − subtract (take away, or "the gap between").0 nothing left over — perfect balance, an empty amount.
Later, many of our sentences will end with "= 0 after combining everything, the two sides balance with nothing left over. It is a tidy way to write a rule, nothing scary.
Keep these four in your pocket — every formula below is built out of them.
Definition Point and coordinatesA point is a single spot on a flat sheet. To name it we use two numbers, written ( x , y )
The first number x right (positive) or left (negative) of the middle.
The second number y up (positive) or down (negative).
The middle spot ( 0 , 0 ) origin .
Look at the figure: the point ( 3 , 2 ) 3 2 address . Every circle we discuss is just a collection of addresses that all obey one rule.
Why the topic needs this: the equation of a circle is a test you apply to an address ( x , y ) on the circle. No coordinate plane, no equation.
A variable is a letter standing for a number we haven't pinned down yet. In a circle equation x y every possible address on the plane, and the equation decides which ones "pass".
Intuition Two jobs for a letterSome letters (like x , y h , k r fixed for one particular circle — they are the circle's identity. Keeping these two jobs separate is the single most important habit in this whole chapter.
The distance rule (next section) needs two operations. We earn them here , before they appear.
2 2 a 2 multiply that thing by itself ": a 2 = a × a 5 2 = 25 ( x − h ) 2 = ( x − h ) × ( x − h ) a 2 area of a square whose side is a ( − 3 ) 2 = 9 = 3 2
Definition The square-root sign a opposite question: "which positive number, when squared, gives what's inside?" So 25 = 5 5 2 = 25 undoes squaring.
One warning we will need later: you cannot take the square root of a negative number and get a real length — no real number squared is negative.
With these two tools defined, we may now build distance.
Before "circle" can mean anything, we need "how far apart are two points?"
Definition Distance between two pointsGiven points ( x 1 , y 1 ) ( x 2 , y 2 ) Δ x = x 2 − x 1 Δ y = y 2 − y 1 d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Why the square root, and why the squares? Look at the red triangle in the figure. The horizontal leg and vertical leg meet at a right angle. The Pythagoras rule says: (long side)2 2 2 undo the squaring and recover the actual slanted length. That square root is the only tool that turns two perpendicular gaps into one straight distance.
Why the topic needs this: a circle is defined as all points at a fixed distance from a centre. Distance is literally the definition machinery.
Pick a fixed point C = ( h , k ) centre , and a fixed positive length r radius . The circle is the set of all points ( x , y ) C r
In the figure, every point on the orange ring is the same distance r inside are closer than r outside are farther. The circle is the exact boundary.
Now write that sentence in symbols using the distance rule from section 4, with ( x 1 , y 1 ) = ( h , k ) ( x 2 , y 2 ) = ( x , y ) ( x − h ) 2 + ( y − k ) 2 = r
Squaring both sides (to get rid of the awkward root — a legal move because both sides are positive) gives the standard form :
( x − h ) 2 + ( y − k ) 2 = r 2
This is the neat, un-disguised sentence. See Standard form of circle for its full treatment.
We already know squaring (section 3). The remaining subtlety here is the minus sign and what it does to the centre.
x − h x + h x − h horizontal gap from the centre's x h x x x − h
Common mistake The sign flip that causes 90% of errorsIn ( x − h ) 2 − h + h ( x − 3 ) 2 x + 3 not − 3 ( − g , − f )
Expand the standard form (multiply everything out). Using ( x − h ) 2 = x 2 − 2 h x + h 2 x 2 − 2 h x + h 2 + y 2 − 2 k y + k 2 = r 2 x 2 + y 2 call 2 g − 2 h x call 2 f − 2 k y + call c ( h 2 + k 2 − r 2 ) = 0
Definition The general form and the lettersg f c general form of a circle:
x 2 + y 2 + 2 g x + 2 f y + c = 0 nicknames for the messy expanded coefficients:
2 g x g = − h 2 f y f = − k c x y h 2 + k 2 − r 2 2 g g 2
It is the same circle as the standard form, just multiplied out and disguised. Recovering h , k , r g , f , c Completing the square .
The radius comes out as r = g 2 + f 2 − c makes sense is decided by comparing the inside to zero, so we need the comparison marks.
Definition Greater-than, equals, less-than
a > b a strictly bigger than b a = b a exactly b a < b a strictly smaller than b > <
Intuition Why the comparison to zero mattersLook at what's under the root, g 2 + f 2 − c
If g 2 + f 2 − c > 0 real circle with a genuine positive radius.
If g 2 + f 2 − c = 0 0 point circle ).
If g 2 + f 2 − c < 0 no real circle (imaginary).
These three comparisons are the gatekeeper that sorts every general-form equation into real, degenerate, or impossible.
Just as important as what's there is what's missing.
Definition Why coefficients ofx 2 y 2 1 x y x 2 y 2 same coefficient (we scale it to 1 x y x y Conic sections general equation .
Coordinate plane and points x,y
Distance between two points
Pythagoras right triangle
Expand and rename to g,f,c
General form to centre and radius
Validity check greater than 0
Every arrow means "you need the left box before the right box makes sense". Notice the topic sits at the very bottom — it is the destination , and every foundation above feeds into it.
Standard form of circle — the tidy sentence we start and end atCompleting the square — the single reversing tool of the whole topicEquation of circle from endpoints of diameter — a common way general form appearsTangent to a circle — builds directly on knowing the centre and radiusFamily of circles — many circles at once, written in general formConic sections general equation — where circles sit among ellipses and hyperbolas↑ Parent topic
Give each a plain answer out loud before revealing.
I know what = + − 0 = + − 0
I can name any point on the plane using two numbers, and I know which is right/left and which is up/down Yes — first number is horizontal (right +), second is vertical (up +); origin is ( 0 , 0 )
I know the difference between a variable (x , y h , k , r Variables roam over all points; h , k , r
I can explain what a 2 a a 2 = a × a ;
a asks "which positive number squared gives
a " — it reverses squaring.
I can state the distance between ( x 1 , y 1 ) ( x 2 , y 2 ) d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 ; the root undoes Pythagoras' squaring to give the true hypotenuse length.
I can define a circle in one sentence using distance All points ( x , y ) ( h , k ) r
I can explain why ( x − 3 ) 2 x + 3 The formula subtracts the coordinate, so the sign inside is the opposite of the coordinate itself.
I can say what g f c g = − h f = − k c = h 2 + k 2 − r 2
I know why x 2 y 2 1 x y A circle is symmetric in all directions; unequal coefficients or an x y
I know what g 2 + f 2 − c > 0 = 0 < 0 > 0 = 0 0 < 0
